An exploration of connections between high school algebra and abstract algebra
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This thesis illuminates some connections between high school algebra and abstract algebra. The concepts of function (e.g., homomorphism and 1-1 correspondence) and algebraic structure (e.g., ring, field, and group) are compatible with each other. This thesis mainly deals with (1) functions from an overall structural perspective, (2) the solvability of some polynomial equations in certain algebraic systems, and (3) the real numbers from both constructive and axiomatic approaches. Throughout this thesis I provide possible questions or tasks that teachers can use to challenge students to think of mathematics in a bigger picture as well as help students develop an admiration for some aesthetic dimensions of mathematics. I identify some questions that high school students might ask and provide answers are given from a relatively advanced mathematical standpoint to illuminate the connections between high school algebra and abstract algebra. The notion of limit in calculus and the interaction between algebra and geometry are also rendered in several contexts.
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